"And thus many are ignorant of mathematical truths, not out of any imperfection of their faculties, or uncertainty in the things themselves, but for want of application in acquiring, examining, and by due ways comparing those ideas." (John Locke, "An Essay Concerning Human Understanding", 1689)
"Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701)
"The mathematics are the friends to religion, inasmuch as they charm the passions, restrain the impetuosity of the imagination, and purge the mind from error and prejudice. Vice is error, confusion and false reasoning; and all truth is more or less opposite to it. Besides, mathematical truth may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices; the delightfulness of them being such as to make solitude not only easy but desirable." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701)
"He that gives a portion of his time and talent to the investigation of mathematical truth will come to all other questions with a decided advantage over his opponents." (Charles C Colton, "Lacon", 1820)
"Each mathematician for himself, and not anyone for any other, not even all for one, must tread that more than royal road which leads to the palace and sanctuary of mathematical truth.” (Sir William R Hamilton, “Report of the Fifth Meeting of the British Association for the Advancement of Science”, [Address] 1835)
"The peculiar character of mathematical truth is that it is necessarily and inevitably true; and one of the most important lessons which we learn from our mathematical studies is a knowledge that there are such truths." (William Whewell, "Principles of English University Education", 1838)
"What is exact about mathematics but exactness? And is not this a consequence of the inner sense of truth?" (Johann Wolfgang von Goethe, "Sprüche in Prosa", 1840)
"Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparisons and measurements. But it was found practicable, and being practicable was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which was obvious to the most careless observer, and which compose the first principles and ultimate premises of the science." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1843)
"He that gives a portion of his time and talent to the investigation of mathematical truth will come to all other questions with a decided advantage over his opponents." (Colton, Charles Caleb, "Lacon; or Many Things in a Few Words", 1849)
"Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but half comprehends the meaning. Experience sees that the assertions are true, but she sees not how profound and absolute is their truth. She unhesitatingly assents to the laws which geometry delivers, but she does not pretend to see the origin of their obligation. She is always ready to acknowledge the sway of pure scientific principles as a matter of fact, but she does not dream of offering her opinion on their authority as a matter of right; still less can she justly claim to herself the source of that authority." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)
"The peculiarity of the evidence of mathematical truths is that all the argument is on one side." (John S Mill, "On Liberty", 1859)
"It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all." (James J Sylvester, "On certain inequalities related to prime numbers", Nature Vol. 38, 1888)
"Mathematics connect themselves on the one side with common life and physical science; on the other side with philosophy in regard to our notions of space and time, and in the questions which have arisen as to the universality and necessity of the truths of mathematics and the foundation of our knowledge of them." (Arthur Cayley, 1888)
"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)
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